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  <h1>Source code for torch._lowrank</h1><div class="highlight"><pre>
<span></span><span class="sd">&quot;&quot;&quot;Implement various linear algebra algorithms for low rank matrices.</span>
<span class="sd">&quot;&quot;&quot;</span>

<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;svd_lowrank&#39;</span><span class="p">,</span> <span class="s1">&#39;pca_lowrank&#39;</span><span class="p">]</span>

<span class="kn">import</span> <span class="nn">torch</span>
<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">_linalg_utils</span> <span class="k">as</span> <span class="n">_utils</span> 
<span class="kn">from</span> <span class="nn">._overrides</span> <span class="kn">import</span> <span class="n">has_torch_function</span><span class="p">,</span> <span class="n">handle_torch_function</span>


<span class="k">def</span> <span class="nf">get_approximate_basis</span><span class="p">(</span><span class="n">A</span><span class="p">,</span>        <span class="c1"># type: Tensor</span>
                          <span class="n">q</span><span class="p">,</span>        <span class="c1"># type: int</span>
                          <span class="n">niter</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>  <span class="c1"># type: Optional[int]</span>
                          <span class="n">M</span><span class="o">=</span><span class="kc">None</span>    <span class="c1"># type: Optional[Tensor]</span>
                          <span class="p">):</span>
    <span class="c1"># type: (...) -&gt; Tensor</span>
    <span class="sd">&quot;&quot;&quot;Return tensor :math:`Q` with :math:`q` orthonormal columns such</span>
<span class="sd">    that :math:`Q Q^H A` approximates :math:`A`. If :math:`M` is</span>
<span class="sd">    specified, then :math:`Q` is such that :math:`Q Q^H (A - M)`</span>
<span class="sd">    approximates :math:`A - M`.</span>

<span class="sd">    .. note:: The implementation is based on the Algorithm 4.4 from</span>
<span class="sd">              Halko et al, 2009.</span>

<span class="sd">    .. note:: For an adequate approximation of a k-rank matrix</span>
<span class="sd">              :math:`A`, where k is not known in advance but could be</span>
<span class="sd">              estimated, the number of :math:`Q` columns, q, can be</span>
<span class="sd">              choosen according to the following criteria: in general,</span>
<span class="sd">              :math:`k &lt;= q &lt;= min(2*k, m, n)`. For large low-rank</span>
<span class="sd">              matrices, take :math:`q = k + 5..10`.  If k is</span>
<span class="sd">              relatively small compared to :math:`min(m, n)`, choosing</span>
<span class="sd">              :math:`q = k + 0..2` may be sufficient.</span>

<span class="sd">    .. note:: To obtain repeatable results, reset the seed for the</span>
<span class="sd">              pseudorandom number generator</span>

<span class="sd">    Arguments::</span>
<span class="sd">        A (Tensor): the input tensor of size :math:`(*, m, n)`</span>

<span class="sd">        q (int): the dimension of subspace spanned by :math:`Q`</span>
<span class="sd">                 columns.</span>

<span class="sd">        niter (int, optional): the number of subspace iterations to</span>
<span class="sd">                               conduct; ``niter`` must be a</span>
<span class="sd">                               nonnegative integer. In most cases, the</span>
<span class="sd">                               default value 2 is more than enough.</span>

<span class="sd">        M (Tensor, optional): the input tensor&#39;s mean of size</span>
<span class="sd">                              :math:`(*, 1, n)`.</span>

<span class="sd">    References::</span>
<span class="sd">        - Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding</span>
<span class="sd">          structure with randomness: probabilistic algorithms for</span>
<span class="sd">          constructing approximate matrix decompositions,</span>
<span class="sd">          arXiv:0909.4061 [math.NA; math.PR], 2009 (available at</span>
<span class="sd">          `arXiv &lt;http://arxiv.org/abs/0909.4061&gt;`_).</span>
<span class="sd">    &quot;&quot;&quot;</span>

    <span class="n">niter</span> <span class="o">=</span> <span class="mi">2</span> <span class="k">if</span> <span class="n">niter</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">niter</span>
    <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span>
    <span class="n">dtype</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">get_floating_dtype</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
    <span class="n">matmul</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">matmul</span>

    <span class="n">R</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">A</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>

    <span class="n">A_H</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transjugate</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
    <span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
        <span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span><span class="o">.</span><span class="n">qr</span><span class="p">()</span>
        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">niter</span><span class="p">):</span>
            <span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A_H</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span><span class="o">.</span><span class="n">qr</span><span class="p">()</span>
            <span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span><span class="o">.</span><span class="n">qr</span><span class="p">()</span>
    <span class="k">else</span><span class="p">:</span>
        <span class="n">M_H</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transjugate</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
        <span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">R</span><span class="p">))</span><span class="o">.</span><span class="n">qr</span><span class="p">()</span>
        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">niter</span><span class="p">):</span>
            <span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A_H</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M_H</span><span class="p">,</span> <span class="n">Q</span><span class="p">))</span><span class="o">.</span><span class="n">qr</span><span class="p">()</span>
            <span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="n">_</span><span class="p">)</span> <span class="o">=</span> <span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">Q</span><span class="p">))</span><span class="o">.</span><span class="n">qr</span><span class="p">()</span>

    <span class="k">return</span> <span class="n">Q</span>


<div class="viewcode-block" id="svd_lowrank"><a class="viewcode-back" href="../../torch.html#torch.svd_lowrank">[docs]</a><span class="k">def</span> <span class="nf">svd_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
    <span class="c1"># type: (Tensor, Optional[int], Optional[int], Optional[Tensor]) -&gt; Tuple[Tensor, Tensor, Tensor]</span>
    <span class="sd">&quot;&quot;&quot;Return the singular value decomposition ``(U, S, V)`` of a matrix,</span>
<span class="sd">    batches of matrices, or a sparse matrix :math:`A` such that</span>
<span class="sd">    :math:`A \approx U diag(S) V^T`. In case :math:`M` is given, then</span>
<span class="sd">    SVD is computed for the matrix :math:`A - M`.</span>

<span class="sd">    .. note:: The implementation is based on the Algorithm 5.1 from</span>
<span class="sd">              Halko et al, 2009.</span>

<span class="sd">    .. note:: To obtain repeatable results, reset the seed for the</span>
<span class="sd">              pseudorandom number generator</span>

<span class="sd">    .. note:: The input is assumed to be a low-rank matrix.</span>

<span class="sd">    .. note:: In general, use the full-rank SVD implementation</span>
<span class="sd">              ``torch.svd`` for dense matrices due to its 10-fold</span>
<span class="sd">              higher performance characteristics. The low-rank SVD</span>
<span class="sd">              will be useful for huge sparse matrices that</span>
<span class="sd">              ``torch.svd`` cannot handle.</span>

<span class="sd">    Arguments::</span>
<span class="sd">        A (Tensor): the input tensor of size :math:`(*, m, n)`</span>

<span class="sd">        q (int, optional): a slightly overestimated rank of A.</span>

<span class="sd">        niter (int, optional): the number of subspace iterations to</span>
<span class="sd">                               conduct; niter must be a nonnegative</span>
<span class="sd">                               integer, and defaults to 2</span>

<span class="sd">        M (Tensor, optional): the input tensor&#39;s mean of size</span>
<span class="sd">                              :math:`(*, 1, n)`.</span>

<span class="sd">    References::</span>
<span class="sd">        - Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding</span>
<span class="sd">          structure with randomness: probabilistic algorithms for</span>
<span class="sd">          constructing approximate matrix decompositions,</span>
<span class="sd">          arXiv:0909.4061 [math.NA; math.PR], 2009 (available at</span>
<span class="sd">          `arXiv &lt;http://arxiv.org/abs/0909.4061&gt;`_).</span>

<span class="sd">    &quot;&quot;&quot;</span>
    <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">jit</span><span class="o">.</span><span class="n">is_scripting</span><span class="p">():</span>
        <span class="n">tensor_ops</span> <span class="o">=</span> <span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span>
        <span class="k">if</span> <span class="p">(</span><span class="ow">not</span> <span class="nb">set</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">type</span><span class="p">,</span> <span class="n">tensor_ops</span><span class="p">))</span><span class="o">.</span><span class="n">issubset</span><span class="p">((</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="kc">None</span><span class="p">)))</span> <span class="ow">and</span> <span class="n">has_torch_function</span><span class="p">(</span><span class="n">tensor_ops</span><span class="p">)):</span>
            <span class="k">return</span> <span class="n">handle_torch_function</span><span class="p">(</span><span class="n">svd_lowrank</span><span class="p">,</span> <span class="n">tensor_ops</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">_svd_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M</span><span class="p">)</span></div>


<span class="k">def</span> <span class="nf">_svd_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
    <span class="c1"># type: (Tensor, Optional[int], Optional[int], Optional[Tensor]) -&gt; Tuple[Tensor, Tensor, Tensor]</span>
    <span class="n">q</span> <span class="o">=</span> <span class="mi">6</span> <span class="k">if</span> <span class="n">q</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">q</span>
    <span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span>
    <span class="n">matmul</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">matmul</span>
    <span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
        <span class="n">M_t</span> <span class="o">=</span> <span class="kc">None</span>
    <span class="k">else</span><span class="p">:</span>
        <span class="n">M_t</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
    <span class="n">A_t</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>

    <span class="c1"># Algorithm 5.1 in Halko et al 2009, slightly modified to reduce</span>
    <span class="c1"># the number conjugate and transpose operations</span>
    <span class="k">if</span> <span class="n">m</span> <span class="o">&lt;</span> <span class="n">n</span><span class="p">:</span>
        <span class="c1"># computing the SVD approximation of a transpose in order to</span>
        <span class="c1"># keep B shape minimal</span>
        <span class="n">Q</span> <span class="o">=</span> <span class="n">get_approximate_basis</span><span class="p">(</span><span class="n">A_t</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M_t</span><span class="p">)</span>
        <span class="n">Q_c</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
        <span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
            <span class="n">B_t</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span>
        <span class="k">else</span><span class="p">:</span>
            <span class="n">B_t</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span>
        <span class="n">U</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">V</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">B_t</span><span class="p">)</span>
        <span class="n">V</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>
    <span class="k">else</span><span class="p">:</span>
        <span class="n">Q</span> <span class="o">=</span> <span class="n">get_approximate_basis</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M</span><span class="p">)</span>
        <span class="n">Q_c</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
        <span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
            <span class="n">B</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A_t</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span>
        <span class="k">else</span><span class="p">:</span>
            <span class="n">B</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A_t</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M_t</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span>
        <span class="n">U</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">V</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">_utils</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">B</span><span class="p">))</span>
        <span class="n">U</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">U</span><span class="p">)</span>

    <span class="k">return</span> <span class="n">U</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">V</span>


<div class="viewcode-block" id="pca_lowrank"><a class="viewcode-back" href="../../torch.html#torch.pca_lowrank">[docs]</a><span class="k">def</span> <span class="nf">pca_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="mi">2</span><span class="p">):</span>
    <span class="c1"># type: (Tensor, Optional[int], bool, int) -&gt; Tuple[Tensor, Tensor, Tensor]</span>
    <span class="sa">r</span><span class="sd">&quot;&quot;&quot;Performs linear Principal Component Analysis (PCA) on a low-rank</span>
<span class="sd">    matrix, batches of such matrices, or sparse matrix.</span>

<span class="sd">    This function returns a namedtuple ``(U, S, V)`` which is the</span>
<span class="sd">    nearly optimal approximation of a singular value decomposition of</span>
<span class="sd">    a centered matrix :math:`A` such that :math:`A = U diag(S) V^T`.</span>

<span class="sd">    .. note:: The relation of ``(U, S, V)`` to PCA is as follows:</span>

<span class="sd">                - :math:`A` is a data matrix with ``m`` samples and</span>
<span class="sd">                  ``n`` features</span>

<span class="sd">                - the :math:`V` columns represent the principal directions</span>

<span class="sd">                - :math:`S ** 2 / (m - 1)` contains the eigenvalues of</span>
<span class="sd">                  :math:`A^T A / (m - 1)` which is the covariance of</span>
<span class="sd">                  ``A`` when ``center=True`` is provided.</span>

<span class="sd">                - ``matmul(A, V[:, :k])`` projects data to the first k</span>
<span class="sd">                  principal components</span>

<span class="sd">    .. note:: Different from the standard SVD, the size of returned</span>
<span class="sd">              matrices depend on the specified rank and q</span>
<span class="sd">              values as follows:</span>

<span class="sd">                - :math:`U` is m x q matrix</span>

<span class="sd">                - :math:`S` is q-vector</span>

<span class="sd">                - :math:`V` is n x q matrix</span>

<span class="sd">    .. note:: To obtain repeatable results, reset the seed for the</span>
<span class="sd">              pseudorandom number generator</span>

<span class="sd">    Arguments:</span>

<span class="sd">        A (Tensor): the input tensor of size :math:`(*, m, n)`</span>

<span class="sd">        q (int, optional): a slightly overestimated rank of</span>
<span class="sd">                           :math:`A`. By default, ``q = min(6, m,</span>
<span class="sd">                           n)``.</span>

<span class="sd">        center (bool, optional): if True, center the input tensor,</span>
<span class="sd">                                 otherwise, assume that the input is</span>
<span class="sd">                                 centered.</span>

<span class="sd">        niter (int, optional): the number of subspace iterations to</span>
<span class="sd">                               conduct; niter must be a nonnegative</span>
<span class="sd">                               integer, and defaults to 2.</span>

<span class="sd">    References::</span>

<span class="sd">        - Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding</span>
<span class="sd">          structure with randomness: probabilistic algorithms for</span>
<span class="sd">          constructing approximate matrix decompositions,</span>
<span class="sd">          arXiv:0909.4061 [math.NA; math.PR], 2009 (available at</span>
<span class="sd">          `arXiv &lt;http://arxiv.org/abs/0909.4061&gt;`_).</span>

<span class="sd">    &quot;&quot;&quot;</span>

    <span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">jit</span><span class="o">.</span><span class="n">is_scripting</span><span class="p">():</span>
        <span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span> <span class="ow">and</span> <span class="n">has_torch_function</span><span class="p">((</span><span class="n">A</span><span class="p">,)):</span>
            <span class="k">return</span> <span class="n">handle_torch_function</span><span class="p">(</span><span class="n">pca_lowrank</span><span class="p">,</span> <span class="p">(</span><span class="n">A</span><span class="p">,),</span> <span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="n">q</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="n">center</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">)</span>

    <span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span>

    <span class="k">if</span> <span class="n">q</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
        <span class="n">q</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
    <span class="k">elif</span> <span class="ow">not</span> <span class="p">(</span><span class="n">q</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">q</span> <span class="o">&lt;=</span> <span class="nb">min</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)):</span>
        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;q(=</span><span class="si">{}</span><span class="s1">) must be non-negative integer&#39;</span>
                         <span class="s1">&#39; and not greater than min(m, n)=</span><span class="si">{}</span><span class="s1">&#39;</span>
                         <span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="nb">min</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
    <span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">niter</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">):</span>
        <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;niter(=</span><span class="si">{}</span><span class="s1">) must be non-negative integer&#39;</span>
                         <span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">niter</span><span class="p">))</span>

    <span class="n">dtype</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">get_floating_dtype</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>

    <span class="k">if</span> <span class="ow">not</span> <span class="n">center</span><span class="p">:</span>
        <span class="k">return</span> <span class="n">_svd_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>

    <span class="k">if</span> <span class="n">_utils</span><span class="o">.</span><span class="n">is_sparse</span><span class="p">(</span><span class="n">A</span><span class="p">):</span>
        <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
            <span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s1">&#39;pca_lowrank input is expected to be 2-dimensional tensor&#39;</span><span class="p">)</span>
        <span class="n">c</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,))</span> <span class="o">/</span> <span class="n">m</span>
        <span class="c1"># reshape c</span>
        <span class="n">column_indices</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">indices</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
        <span class="n">indices</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">column_indices</span><span class="p">),</span>
                              <span class="n">dtype</span><span class="o">=</span><span class="n">column_indices</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span>
                              <span class="n">device</span><span class="o">=</span><span class="n">column_indices</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
        <span class="n">indices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">column_indices</span>
        <span class="n">C_t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sparse_coo_tensor</span><span class="p">(</span>
            <span class="n">indices</span><span class="p">,</span> <span class="n">c</span><span class="o">.</span><span class="n">values</span><span class="p">(),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">A</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>

        <span class="n">ones_m1_t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">A</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
        <span class="n">M</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">mm</span><span class="p">(</span><span class="n">C_t</span><span class="p">,</span> <span class="n">ones_m1_t</span><span class="p">))</span>
        <span class="k">return</span> <span class="n">_svd_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M</span><span class="p">)</span>
    <span class="k">else</span><span class="p">:</span>
        <span class="n">c</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,))</span> <span class="o">/</span> <span class="n">m</span>
        <span class="n">C</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
        <span class="n">ones_m1</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">A</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
        <span class="n">M</span> <span class="o">=</span> <span class="n">ones_m1</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
        <span class="k">return</span> <span class="n">_svd_lowrank</span><span class="p">(</span><span class="n">A</span> <span class="o">-</span> <span class="n">M</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span></div>
</pre></div>

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